Oscillating Turing patterns, chaos and strange attractors in a reaction-diffusion system augmented with self-and cross-diffusion terms

In this article we introduce an original model in order to study the emergence of chaos in a reaction diffusion system in the presence of self-and cross-diffusion terms. A Fourier Spectral Method is derived to approximate equilibria and orbits of the latter. Special attention is paid to accuracy, a necessary condition when one wants to catch periodic orbits and to perform their linear stability analysis via Floquet multipliers. Bifurcations with respect to a single control parameter are studied in four different regimes of diffusion: linear diffusion, self-diffusion for each of the two species, and cross-diffusion. Key observations are made: development of original Turing patterns, Hopf bifurcations leading to oscillating patterns and period doubling cascades leading to chaos. Eventually, original strange attractors are reported in phase space.